三维及更高维度几何相交图直径计算景观的绘制
Charting the Diameter Computation Landscape of Geometric Intersection Graphs in Three Dimensions and Higher
AI总结：
本文研究了三维及更高维度几何相交图直径计算问题，提出首个三维对象直径3的亚二次时间算法，并给出单位球体在直径3下的亚二次时间下界，同时扩展了矩形框直径2和3的算法与下界。
摘要：
最近关于计算几何相交图直径的研究取得了显著进展，主要集中在2D情况，为简单对象如单位圆和轴对齐正方形提供了真正的亚二次时间算法。然而，在三维或更高维度中，对于任何非平凡对象的相交图，尚无已知的真正亚二次时间算法，即使对于基本对象如单位球或轴对齐单位立方体也是如此。这一现象部分由Bringmann等人[SoCG '22]的开创性工作解释，他们给出了几个真正的亚二次时间下界，特别是在三维中当图直径Δ≥Ω(log n)时，单位球或单位立方体的下界，暗示了更高维度直径问题的复杂性悲观前景。在本文中，我们大幅扩展了三维及更高维度对象直径计算的景观，给出了几个积极结果。我们的主要发现包括：- 一个确定三维单位立方体直径至多3的真正亚二次时间算法（Diameter-3在此后简称），这是首个针对三维或更高维度对象的此类算法。该算法基于一种新的伪线联系，具有独立兴趣。- 在正交向量（OV）假设下，对于三维单位球体的Diameter-3给出真正的亚二次时间下界，首次在小直径范围内区分单位球体和单位立方体。此前，当直径为Ω(log n)时，计算这两种对象的直径已知为真正的亚二次时间困难。- 一个近线性时间算法用于三维单位立方体的Diameter-2，推广了之前二维单位正方形的结果。- 对于任意维度和大小的矩形框，分别给出Diameter-2和Diameter-3的真正亚二次时间算法和下界。
英文摘要：
Recent research on computing the diameter of geometric intersection graphs has made significant strides, primarily focusing on the 2D case where truly subquadratic-time algorithms were given for simple objects such as unit-disks and (axis-aligned) squares. However, in three or higher dimensions, there is no known truly subquadratic-time algorithm for any intersection graph of non-trivial objects, even basic ones such as unit balls or (axis-aligned) unit cubes. This was partially explained by the pioneering work of Bringmann et al. [SoCG '22] which gave several truly subquadratic lower bounds, notably for unit balls or unit cubes in 3D when the graph diameter $Δ$ is at least $Ω(\log n)$, hinting at a pessimistic outlook for the complexity of the diameter problem in higher dimensions. In this paper, we substantially extend the landscape of diameter computation for objects in three and higher dimensions, giving a few positive results. Our highlighted findings include: - A truly subquadratic-time algorithm for deciding if the diameter of unit cubes in 3D is at most 3 (Diameter-3 hereafter), the first algorithm of its kind for objects in 3D or higher dimensions. Our algorithm is based on a novel connection to pseudolines, which is of independent interest. - A truly subquadratic time lower bound for \Diameter-3 of unit balls in 3D under the Orthogonal Vector (OV) hypothesis, giving the first separation between unit balls and unit cubes in the small diameter regime. Previously, computing the diameter for both objects was known to be truly subquadratic hard when the diameter is $Ω(\log n)$. - A near-linear-time algorithm for Diameter-2 of unit cubes in 3D, generalizing the previous result for unit squares in 2D. - A truly subquadratic-time algorithm and lower bound for Diameter-2 and Diameter-3 of rectangular boxes (of arbitrary dimension and sizes), respectively.
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